Replace the numerals 1 through 8 with ever increasing prime numbers, always using the next lowest possible that is available1 to fulfill the criteria on the left. Then do the same for the right.

Present a series for the left, and one for the right.

7 + 8 = Square

6 = Prime

4 + 5 = Square

1 + 2 + 3 = Square

7 + 8 = Cube

6 = Prime

4 + 5 = Cube

1 + 2 + 3 = Cube

If it was required that the Right set required "1" to be the next Prime following on after that used for the "8" in the Left set, what might the Right set read, if indeed it is possible?

1.
Note, "always using the next lowest possible that is available" means that if it is next on the list it cannot be dismissed unless it is the last of a group of two or three and will not fulfill the criterion. Only then may you advance to the next.